On the best approximations and rate of convergence of decompositions in the root vectors of an operator
We establish upper bounds of the best approximations of elements of a Banach space B by the root vectors of an operator $A$ that acts in B. The corresponding estimates of the best approximations are expressed in terms of a K-functional associated with the operator A. For the operator of differentiat...
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| Datum: | 1997 |
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| Format: | Artikel |
| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
1997
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/5060 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860511256823201792 |
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| author | Radzievskii, G. V. Радзиевский, Г. В. Радзиевский, Г. В. |
| author_facet | Radzievskii, G. V. Радзиевский, Г. В. Радзиевский, Г. В. |
| author_sort | Radzievskii, G. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:24:02Z |
| description | We establish upper bounds of the best approximations of elements of a Banach space B by the root vectors of an operator $A$ that acts in B. The corresponding estimates of the best approximations are expressed in terms of a K-functional associated with the operator A. For the operator of differentiation with periodic boundary conditions, these estimates coincide with the classical Jackson inequalities for the best approximations of functions by trigonometric polynomials. In terms of K-functionals, we also prove the abstract Dini-Lipschitz criterion of convergence of partial sums of the decomposition of f from B in the root vectors of the operator A to f |
| first_indexed | 2026-03-24T03:10:00Z |
| format | Article |
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| id | umjimathkievua-article-5060 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:10:00Z |
| publishDate | 1997 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b5/0463b77e52ee3210550995fcc7b9ceb5.pdf |
| spelling | umjimathkievua-article-50602020-03-18T21:24:02Z On the best approximations and rate of convergence of decompositions in the root vectors of an operator О наилучших приближениях и о скорости сходимости разложений по корневым векторам оператора Radzievskii, G. V. Радзиевский, Г. В. Радзиевский, Г. В. We establish upper bounds of the best approximations of elements of a Banach space B by the root vectors of an operator $A$ that acts in B. The corresponding estimates of the best approximations are expressed in terms of a K-functional associated with the operator A. For the operator of differentiation with periodic boundary conditions, these estimates coincide with the classical Jackson inequalities for the best approximations of functions by trigonometric polynomials. In terms of K-functionals, we also prove the abstract Dini-Lipschitz criterion of convergence of partial sums of the decomposition of f from B in the root vectors of the operator A to f Отримано оцінки зверху найкращих наближень елементів банахового простору B за допомогою кореневих векторів оператора $A$, що діє в B. Відповідні оцінки найкращих наближень знайдено у термінах K-функціонала, який побудовано за оператором $A$. Для оператора диференціювання з періодичною крайовою умовою ці оцінки збігаються з класичними нерівностями Джексона про оцінки найкращих наближень функції за допомогою тригонометричнів поліномів. У термінах K-функціоналів доведена також абстрактна ознака Діні — Ліпшіца про збіжність частинних сум розкладу f з B за кореневими векторами оператора А до f. Institute of Mathematics, NAS of Ukraine 1997-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5060 Ukrains’kyi Matematychnyi Zhurnal; Vol. 49 No. 6 (1997); 754–773 Український математичний журнал; Том 49 № 6 (1997); 754–773 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/5060/6817 https://umj.imath.kiev.ua/index.php/umj/article/view/5060/6818 Copyright (c) 1997 Radzievskii G. V. |
| spellingShingle | Radzievskii, G. V. Радзиевский, Г. В. Радзиевский, Г. В. On the best approximations and rate of convergence of decompositions in the root vectors of an operator |
| title | On the best approximations and rate of convergence of decompositions in the root vectors of an operator |
| title_alt | О наилучших приближениях и о скорости сходимости разложений по корневым векторам оператора |
| title_full | On the best approximations and rate of convergence of decompositions in the root vectors of an operator |
| title_fullStr | On the best approximations and rate of convergence of decompositions in the root vectors of an operator |
| title_full_unstemmed | On the best approximations and rate of convergence of decompositions in the root vectors of an operator |
| title_short | On the best approximations and rate of convergence of decompositions in the root vectors of an operator |
| title_sort | on the best approximations and rate of convergence of decompositions in the root vectors of an operator |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5060 |
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